Like I mentioned once already
What size center are you using to mark the stock? You are saying that the through hole is already drilled! What diameter? That is itself may be the problem! It's not that the cemter pin does not stick out far enough beyond the ring, the center hole is larger than the pin to begin with!!
Try marking an X on the stock ends. Then make a center punch mark where the lines intersect.....THEN try your live center!
I do it this way. Then when I drill I go to a regular conical live center to hold the plug body (no real tailstock pressure required.)